About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 427393, 12 pages
http://dx.doi.org/10.1155/2012/427393
Research Article

Application of Multistep Generalized Differential Transform Method for the Solutions of the Fractional-Order Chua's System

Department of Mathematics, Faculty of Science, The University of Jordan, Amman 1194, Jordan

Received 29 July 2012; Accepted 18 October 2012

Academic Editor: Mingshu Peng

Copyright © 2012 Asad Freihat and Shaher Momani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus; Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH
  3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH
  4. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH
  5. C. Li and W. Deng, “Remarks on fractional derivatives,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 777–784, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, Singapore, 2009.
  8. R. L. Bagley and P. J. Torvik, “A theoretical basis for the application of fractional calculus,” Journal of Rheology, vol. 27, pp. 201–210, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. R. L. Bagley and R. A. Calico, “Fractional order state equations for the control of viscoelastically damped structures,” Journal of Guidance, Control, and Dynamics, vol. 14, no. 2, pp. 304–311, 1991. View at Publisher · View at Google Scholar
  10. E. J. S. Pires, J. A. T. Machado, and P. B. de Moura, “Fractional order dynamics in a GA planner,” Signal Processing, vol. 83, pp. 2377–2386, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. K. S. Hedrih and V. N. Stanojević, “A model of gear transmission: fractional order system dynamics,” Mathematical Problems in Engineering, vol. 2010, Article ID 972873, 2010.
  12. J. Cao, C. Ma, H. Xie, and Z. Jiang, “Nonlinear dynamics of duffing system with fractional order damping,” Journal of Computational and Nonlinear Dynamics, vol. 5, no. 4, pp. 041012–041018, 2010. View at Publisher · View at Google Scholar
  13. C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar
  14. K. Moaddy, S. Momani, and I. Hashim, “The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1209–1216, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. A. G. Radwan, K. Moaddy, and S. Momani, “Stability and non-standard finite difference method of the generalized Chua's circuit,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 961–970, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. I. Petrás, “A note on the fractional-order Chua's system,” Chaos Solution & Fractals, vol. 38, pp. 140–147, 2008. View at Publisher · View at Google Scholar
  17. T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos on a fractional Chua's system,” IEEE Transactions on Circuits and Systems, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar
  18. Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467–477, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 85–95, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1462–1472, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. V. S. Ertürk, Z. M. Odibat, and S. Momani, “An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 996–1002, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. V. Erturk, G. Zaman, and S. Momani, “A numericanalytic method for approximating a giving up smoking model containing fractional derivatives,” Computers & Mathematics With Applications, vol. 64, no. 10, pp. 3065–3074, 2012. View at Publisher · View at Google Scholar